[Physics.Gen-Ph] 1 Dec 2019 the Benefits of Affine Quantization

Home , Quartic interaction

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2 Classical Variable Connections

2.1 Traditional classical variables We ﬁrst focus on a single degree of freedom. The familiar integral that leads to the equations of motion is derived from an action functional (with p = p(t) and q = q(t) , 0 ≤ t ≤ T > 0 ) given by

T A = R 0 [p q˙ − H(p, q)] dt . (1) Standard stationary relations, including δp(0)=0= δp(T ) and δq(0)=0= δq(T ), such that

T δA = R 0 {(q ˙ − ∂H(p, q)/∂p ) δp − (p ˙ + ∂H(p, q)/∂q ) δq} dt =0 , (2) leads to two, basic, equations of motion,

q˙ = ∂H(p, q)/∂p , p˙ = −∂H(p, q)/∂q . (3)

These equations of motion apply whatever the range of the basic variables, p and q. As examples of interest, let us assume (1) −∞

2 2.2 Traditional aﬃne variables New, and speciﬁc, phase-space coordinates may be introduced to Eq. (1) by the relation p dq = rds, and we choose r ≡ p q and s ≡ ln(q). Note that the choice of s implies that 0

T ′ A = R 0 [rs˙ − H (r, s)] dt (4) where H′(r, s) ≡ H(p, q). We next introduce stationary relations

T ′ ′ δA = R 0 {(s ˙ − ∂H (r, s)/∂r ) δr − (r ˙ + ∂H (r, s)/∂s ) δs} dt =0 , (5) where δr(0) = 0 = δs(0) as well as δr(T )=0= δs(T ), which leads to two equations of motion given by

s˙ = ∂H′(r, s)/∂r , r˙ = −∂H′(r, s)/∂s . (6)

3 Quantum Operator Connections

3.1 Aﬃne quantum operators A conventional canonical quantization promotes classical variables to quantum operators such as p → P and q → Q, and r → R and s → S. These operators satisfy [Q, P ]= i~11and [S, R]= i~11as well. But, since s = ln(q) or q = es, there should be some sort of connection between Qand S, such as [Q, S] = 0, and S = ln(Q) and, therefore, Q = eS. We can support these equations by diagonalizing these variables. In particular, we have the states |qi, 0

[Sn, R]= i~ nSn−1 ∞ ∞ n n−1 X(1/n!) [S , R]= i~ X(n/n!) S n=1 n=1

3 [eS − 1, R]= i~ eS [eS, R]= i~ eS [Q, R]= i~ Q, (7) When 0

3.2 Coherent states Coherent states provide a connection between the classical and the quantum realms. Simple coherent states rely on two classical variables and two quantum operators, the last of which are self-adjoint operators. For systems for which −∞

4 3.3 Quantum/classical transition 3.3.1 the canonical story The quantum action functional leading to Schr¨odinger’s equation is given, for normalized Hilbert space vectors |ψ(t)i, by T ~ A = R 0 hψ(t)|[i ∂/∂t − H(P, Q)] |ψ(t)i dt . (10) A general stationary variation leads to Schr¨odinger’s equation,

i~ ∂|ψ(t)i/∂t = H(P, Q) |ψ(t)i (11) and its adjoint. However, macroscopic observers can only vary a limited set of vectors, such as the coherent states. This leads to a reduced (r) action functional given by T ~ Ar = R 0 hp(t), q(t)|[i ∂/∂t − H(P, Q)] |p(t), q(t)i dt T = R 0 [p(t)q ˙(t) − H(p(t), q(t)] dt , (12) and the ﬁnal result appears to be a classical action functional, and, indeed, one that has an ‘advantage’ because ~ > 0 just as it is in the real world! To complete this story we examine the quantum Hamiltonian, H(P, Q), and the ‘enhanced’ (because ~ > 0) classical Hamiltonian, H(p, q), and we learn that

H(p, q) = hp.q|H(P, Q)|p, qi = h0|H(P + p, Q + q)|0i = H(p, q)+ O(~; p, q) . (13)

It follows that in the true classical limit, where ~ → 0 and H(p, q) → Hc(p, q) (c for classical), the quantum operators assume the positions of the classical variables in the Hamiltonian. Moreover, the favored phase-space coordinates that lead to a physically correct quantization, as Dirac observed [6], are Cartesian coordinates. This proposal is conﬁrmed [1] by

dσ(p, q)2 ≡ 2~ [|| d|p, qi ||2 − |hp, q| d|p, qi|2]= ω−1dp2 + ω dq2 . (14)

These phase-space variables are the favored ones to promote to quantum operators to have a physically correct quantization. For a comparison purpose,

5 which is coming up, we note that the two-dimensional ﬂat space in (14) is a constant zero curvature space. The foregoing material was devoted to situations where −∞

3.3.2 the affine story The quantum action functional for affine variables begins much as the canonical variable story, again for normalized Hilbert states |ψ(t)i. In the new situation (with a prime to recognize R in place of P ) we have T ~ ′ A = R 0 hψ(t)|[i ∂/∂t − H (R, Q)]|ψ(t)i dt , (15) which, with stationary variations, leads directly to Schrödinger’s equation and its adjoint. Restricting the variations to affine coherent states leads to T ~ ′ Ar = R 0 hp(t); q(t)|[i ∂/∂t − H (R, Q)]|p(t); q(t)i dt T = R 0 [−q(t)p ˙(t) − H(p(t), q(t))] dt . (16) Recalling that in this case both q and Q are dimensionless, it follows that

H(p, q) = hp; q|H′(R, Q)|p; qi = hβ|H′(R + pqQ, qQ)|βi = H′(pq,q)+ O(~; p, q) = H(p, q)+ O(~; p, q) . (17)

Once again we see that H(p, q)= Hc(p, q) in the classical limit when ~ → 0. Are the favored phase-space variables Cartesian coordinates in the aﬃne case? The answer is no. In particular, we ﬁnd [1] that

dσ(p, q)2 ≡ 2~[|| d|p; qi ||2 − |hp; q| d|p; qi|2]= β−1q2 dp2 + β q−2 dq2 . (18)

This two-dimensional space is not ﬂat, but it is a constant negative curvature space, with a negative curvature that is −2/β. In the sense of having a constant curvature value this space is in the same category as the ﬂat space. These phase-space variables are the favored ones to promote to quantum operators in order to have a physically correct quantization.3

3Unlike a ﬂat plane, or a constant positive curvature surface (which holds the metric

6 3.3.3 the classical/quantum story Observe that the classical story, as derived from the proper quantum operators and the appropriate coherent states in (12) and (16), lead to a very similar classical story. So similar are their stories that if one of the quantizing procedures fails then there is a very good chance that the other quantization procedure could succeed. This similarity enables us to claim that affine quantization should be con- sidered as a parallel road that starts in phase space and results in an accept- able quantum theory for selected problems, which is exactly what canonical quantization has to offer for a different set of selected problems. The union of canonical quantization and affine quantization is called ‘enhanced quantization’, and this program can be found in [5]. Hereafter, we examine three classical systems that canonical quantization fails to solve but affine quantization is able to solve. These examples include a single degree-of-freedom model, and two field theories: one about quartic scalar fields in 4, and more, spacetime dimensions, and the other is Einstein’s theory of gravity in general relativity in 4 spacetime dimensions.4

4 The “Harmonic Oscillator”

The standard harmonic oscillator entails −∞ < p,q < ∞ with a classical Hamiltonian H(p, q)=(p2 + q2)/2. Its canonical quantization is well known and need not be repeated here. However, our example (which gives rise to the quotation marks in this section’s title) has the same Hamiltonian, H(p, q)=(p2 + q2)/2, with −∞

H(p, q)=(p2 + q2)/2=(pq q−2 pq + q2)/2=(r q−2 r + q2)/2= H′(r, q).(19) of three-dimensional spin coherent states), a space of constant negative curvature can not be visualized in a 3-dimensional flat space [7]. At every point in this space the negative curvature appears like a saddle having an ‘up curve’ in the direction of the rider’s chest and a ‘down curve’ in the direction of the rider’s legs. 4Path integrals are commonly used to quantize classical theories as well. A mathe- matically sound procedure, which uses different sets of coherent states so as to feature specific quantizations, either a canonical quantization, an affine quantization, or a spin quantization, is discussed in [8], Chap. 8.

7 According to (17), we choose the quantum Hamiltonian to be

H′(R, Q)=(R Q−2 R + Q2)/2 , (20) and according to (7) we have [Q, R] = i~ Q. Adopting a Schr¨odinger repre- 1 ~ sentation for Q = x, with 0

~ 1 1 ~2 −2 i ∂ ψ(x, t)/∂t = 2 {− 4 [x(∂/∂x)+(∂/∂x)x] x ×[x(∂/∂x)+(∂/∂x)x]+ x2} ψ(x, t) 1 ~2 2 2 3 ~2 −2 2 = 2 [− ∂ /∂x + 4 x + x ] ψ(x, t) . (21) This latter problem has also been examined using canonical quantization, an analysis which fails. That story can be found in [1], Sec. 1.5.

5 Scalar Quartic Interaction Field Theories in High Spacetime Dimensions

Quartic interactions for scalar field theories may have satisfactory canonical quantizations provided the spacetime dimension n ≤ 3. For n = 4 canonical quantization leads to a free quantum theory, despite the coupling constant go > 0 [9], and for n ≥ 5, the analysis becomes nonrenormalizable. The classical version of such models is able to avoid high energy regions provided by the interaction term. For a finite overall energy the classical behavior of these models is well behaved. However, quantization can not avoid the domain issues that exist. Specifically, assume that the domain for a free scalar field φ(x) is Dgo=0, but when the interaction is introduced, i.e., go > 0, then the domain of the scalar field is immediately reduced, i.e.,

Dgo>0 ⊂ Dgo=0. The new, smaller domain survives even when go → 0. This behavior is not significant for classical solutions, but such domain behavior complicates the quantization of such models. Since canonical quantization can not quantize them, let us try affine quantization. It is newsworthy that these models can be solved via affine quantization, but that solution is not dependent on ‘positivity restrictions’ for fields such as 0 < φ(x) < ∞. Instead, affine quantization leads to positive results simply by combining a momentum field π(x) and a scalar field φ(x) =6 0, i.e., κ(x) = π(x) φ(x), and using κ(x) instead of π(x). The secret of affine

8 quantization lies in being ready to accept a new ground state dictated by the reduction of domains.5

5.1 Quantization of selected scalar field theories In this section we examine a scalar field with a quartic interaction and we employ an affine quantization. The model has a standard classical Hamilto- nian,

1 2 −→ 2 2 2 4 s H(π,φ)= R { 2 [π(x) +(∇φ(x)) + mo φ(x) ]+ go φ(x) } d x ; (22) here s denotes the number of spatial coordinates. Next, we introduce the aﬃne ﬁeld κ(x) ≡ π(x) φ(x), and modify the classical Hamiltonian to become

′ 1 −2 −→ 2 2 2 H (κ, φ)= R { 2 [κ(x)φ(x) κ(x)+(∇φ(x)) + mo φ(x) ] 4 s +go φ(x) } d x . (23)

For quantization it follows that φ(x) → φˆ(x) =6 0 plus κ(x) → κˆ(x), and we accept that [φˆ(x), κˆ(y)] = i~ δ(x − y) φˆ(x). Additionally, we assume we have a Schr¨odinger representation in which φˆ(x)= φ(x) and

1 ~ κˆ(x)= − 2 i [φ(x)(δ/δφ(x))+(δ/δφ(x))φ(x)] . (24) It follows that Schr¨odinger’s equation is given by

~ 1 −2 −→ 2 2 2 i ∂ Ψ(φ, t)/∂t = R { 2 [ˆκ(x) φ(x) κˆ(x)+(∇φ(x)) + mo φ(x) ] 4 s +go φ(x) } d x Ψ(φ, t) . (25)

−1/2 −1/2 It is noteworthy thatκ ˆ(x) φ(x) =0, as well asκ ˆ(x) Πy φ(y) = 0. Solutions of this Schrödinger equation may be made simpler if its continuum formulation is first replaced by a suitable lattice regularization to facilitate finding solutions, and then, possibly, followed by an appropriate continuum limit; see [4]. There are some limited Monte Carlo studies of affine quantization of these scalar models; additional studies are welcome.

5An aﬃne quantization for an ultralocal (meaning: no gradients) scalar model in [4] illustrates how the ground state for this model is not the usual free ground state. Such behavior is referred to as ‘pseudofree’.

9 6 Quantum Gravity

The basic variables for classical gravity are the symmetric-index metric gab(x), a, b, ··· = 1, 2, 3, and the matrix {gab(x)} > 0 which implies that g(x) ≡ det[gab(x)] > 0 as well. In addition, the symmetric-index momentum is given by πcd(x), which are arbitrary real variables. a ac An important combination is πb (x) ≡ π (x) gbc(x), with a summation over c. We name this combination term ‘momentric’ because it involves the momentum and the metric. To obtain the affine field we should combine the momentum and metric fields, which points us to the momentric field even though there is a difference in the number of independent metric terms (six) and momentric terms (nine). The Poisson brackets for these fields leads to

a c ′ 1 3 ′ a c c a {πb (x), πd(x )} = 2 δ (x − x )[δd πb (x) − δb πd (x)] , c ′ 1 3 ′ c c {gab(x), πd(x )} = 2 δ (x − x )[δagbd(x)+ δbgad(x)] , (26) ′ {gab(x),gcd(x )} =0 .

Unlike the canonical Poisson brackets, these Poisson brackets suggest that they are equally valid if gab(x) → −gab(x), and thus there could be separate realizations of each choice.6 Passing to operator commutations, we are led by (17) and suitable coherent states [1], to promote the Poisson brackets to the operators

a c ′ 1 ~ 3 ′ a c c a [ˆπb (x), πˆd(x )] = i 2 δ (x − x )[δd πˆb (x) − δb πˆd (x)] , c ′ 1 ~ 3 ′ c c [ˆgab(x), πˆd(x )] = i 2 δ (x − x )[δagˆbd(x)+ δbgˆad(x)] , (27) ′ [ˆgab(x), gˆcd(x )]=0 .

There are two irreducible representations of the metric tensor operator con- sistent with these commutations: one where the matrix {gˆab(x)} > 0, which we accept, and one where the matrix {gˆab(x)} < 0, which we reject. The classical Hamiltonian for our models is given [11] by

−1/2 a b 1 a b 1/2 (3) 3 H(π,g)= R {g(x) [πb (x)πa(x) − 2 πa (x)πb(x)] + g(x) R(x)} d x, (28) where (3)R(x) is the 3-dimensional Ricci scalar. For the quantum operators we adopt a Schr¨odinger representation for the basic operators: speciﬁcally

6An earlier analysis of these issues appears in [10].

10 gˆab(x)= gab(x) and a 1 ~ πˆb (x)= − 2 i [ gbc(x)(δ/δgac(x))+(δ/δgac(x))) gbc(x)] . (29) It follows that the Schr¨odinger equation is given by

~ a −1/2 b 1 a −1/2 b i ∂ Ψ({g}, t)/∂t)= R {[ˆπb (x) g(x) πˆa(x) − 2 πˆa(x) g(x) πˆb(x)] +g(x)1/2 (3)R(x)} d3x Ψ({g}, t) , (30) where {g} represents the gab(x) matrix. It is noteworthy [4] that

a −1/2 a −1/2 πˆb (x) g(x) =0 , πˆb (x) Πy g(y) =0 . (31) Indeed, these two equations imply that the factor g(x)−1/2 can be moved to the left in the Hamiltonian in (30). Using that fact we can change the Hamiltonian operator, essentially by multiplying the Hamiltonian by g(x)1/2, and using that expression to make the result a simpler approach to fulﬁll the Hamiltonian constraints [11] to seek Hilbert space states Ω({g}) such that

a b 1 a b (3) {[ˆπb (x)ˆπa(x) − 2 πˆa (x)ˆπb(x)] + g(x) R(x)} Ω({g})=0 . (32) The analysis this far deals with the most diﬃcult hurdle in the quantization of gravity. Further studies in this story are available in [1] as well as [4].

7 Overview and Completeness

Let us begin by returning to a single degree of freedom, like p and q, in the connection of canonical and aﬃne procedures. The common action functional for canonical procedures is given by

T A = R 0 {p q˙ − H(p, q) } dt . (33) As we presented earlier, stationary variations lead to the classical equations of motion. Selection of special phase-space variables, namely the ‘Cartesian coordinates’ [1], say p and q, should be promoted to quantum operators, namely P and Q, which obey [Q, P ]= i~11. Reduced to these few sentences, we have captured the essence of canonical quantization. We now turn to aﬃne quantization.

11 We modify the classical action functional as follows by featuring a new variable r ≡ p q in place of p. This change leads to

T ′ A = R 0 { r q/q˙ − H (r, q) } dt . (34) Stationary variations again lead to the relevant equations of motion. Observe that the termq/q ˙ can deal with 0

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